eliminate the harmonic current and therefore reduce the filter grid side inductor. However ... Simulation results are presented and discussed in order to prove the efficiency ... injection into the grid. ... terms as well as robustness against external disturbances and ..... of the outer dc-link voltage control loop, whereas the q-axis.
INTERNATIONAL JOURNAL of RENEWABLE ENERGY RESEARCH M. Ben Saïd-Romdhane et al., Vol.7, No.3, 2017
Indirect Sliding Mode Power Control associated to Virtual-Resistor-based Active Damping Method for LLCL-Filter-based Grid-Connected Converters M. Ben Saïd-Romdhane*, M. W. Naouar**, I. Slama-Belkhodja***, E. Monmasson**** *,**,***Université de Tunis El Manar, Ecole Nationale d'Ingénieurs de Tunis, LR11ES15 Laboratoire de Systèmes Electriques, 1002, Tunis, Tunisie ****SATIE, University of Cergy-Pontoise, 33 bd du Port, 95000 Cergy-Pontoise, France Received: 14.01.2017 Accepted:03.04.2017 Abstract- LLCL filters are becoming an attractive solution for Grid connected Converters (GcCs) due to their ability to reduce the filter cost and size while meeting new grid codes and power quality requirements. Compared to the conventional LCL filter, the LLCL filter introduces an additional inductor in the capacitor branch to form a series LC circuit that resonates at the GcC switching frequency. The LC branch has a low impedance at the switching frequency, which can greatly eliminate the harmonic current and therefore reduce the filter grid side inductor. However, the LLCL filter resonance phenomenon and the large changes in the grid inductance (typically under weak grid conditions and in rural areas) may compromise the system stability. In order to address these concerns, this paper proposes an Indirect Sliding Mode Power Control associated to Virtual Resistor based Active Damping method (ISMPC-VRAD) for LLCL-filter-based Gridconnected Converters (LLCL-GcCs). The LLCL filter design parameters as well as the ISMPC-VRAD gains are carefully tuned in order to ensure stable operation under severe grid inductance variations while taking into account the influence of LLCL filter parameters changes on the system stability. Simulation results are presented and discussed in order to prove the efficiency and the reliability of the proposed ISMPC-VRAD for LLCL-GcCs as well as high filtering performances of the designed LLCL-GcCs. Keywords LLCL filter, Grid connected Converters (GcCs), Indirect Sliding Mode Power Control (ISMPC), Virtual Resistor based Active Damping method (VRAD), large grid impedance variation. Nomenclature AD GcC ISMPC LLCL-GcC PCC PD THD VRAD SVM Cf fg fres fsω ic(a,b,c) idq ii(a,b,c) iin i2n Isat isω i2(a,b,c) i* Kr Lf Lg Li LT L2
Active Damping Grid connected Converter Indirect Sliding Mode Power Control LLCL-filter-based Grid-connected Converters Point of common coupling Passive Damping Total Harmonic Distortion Virtual Resistor based Active Damping method Space Vector Modulation LLCL filter capacitance rated frequency of grid voltage resonance frequency of LLCL filter switching frequency of the power converter LLCL filter capacitor current (for k=a,b,c phases) dq-axis current of dq transformation converter side current (for k=a,b,c phases) n harmonic order of the converter side current n harmonic order of the grid side current saturation current of the LCLL filter inductors switching frequency harmonic order of the current grid side current (for k=a,b,c phases) reference current virtual resistor trap inductance of the LLCL filter inductive part of the grid impedance LLCL filter converter side inductance LLCL filter total inductance LLCL filter grid side inductance
L2g P Pn Q Qc Rch Rd Si(a,b,c) Si2dq ̇ Ugn Vc(a,b,c) V’ c Vdc Vdq Vdqatt Vdqeq Vg(a,b,c) Vi(a,b,c) Vin V* θdq δ Δimax ωg ωres ωsω
sum of filter filter side inductance and grid inductance active power rated active power reactive power reactive power consumed by the filter capacitor LLCL filter resistive load damping resistance switching states of the converter (for k=a,b,c phases) switching functions time derivatives of the switching functions line-to-line RMS grid voltage voltage across the filter capacitor in series with the damping resistor (for k=a,b,c phases) voltage across the filter capacitor voltage of power converter capacitor dq-axis voltage of dq transformation attractive voltage vector equivalent voltage vector grid voltage (for k=a,b,c phases) power converter output voltage (for k=a,b,c phases) n harmonic order of the converter side voltage reference voltage grid voltage vector phase harmonic attenuation rate maximum converter current ripple rated angular frequency of the grid voltage resonance angular frequency of the filter switching angular frequency of the converter
1. Introduction In order to meet new grid codes and their on-going changes in the near future, Grid-connected Converters (GcCs) employ a passive low-pass filter for connection with the grid. The introduced filter can be either an L filter (Fig.1.a), an LCL filter (Fig.1.b), an LLCL filter (Fig.1.c) or a multi-tuned filter (Fig.1.d). Despite being simple, the L filter has bulky size (large inductance value) and low harmonic attenuation characteristic. Compared with L filter, the LCL filter is a third order system able to reduce significantly switching harmonic injection as well as filter cost and encumbrance [1][2][3]. The required inductor and capacitor values can further be reduced by replacing the capacitor branch of an LCL filter with a series LC trap, as shown in Fig.1.c. The resulting filter is named LLCL filter [4]. Compared to the LCL filter, the LLCL filter has an extra small inductor Lf added in series with the filter capacitor.
This inductor, together with the capacitor Cf, is tuned to resonate at the GcC switching frequency. Harmonics around the switching frequency will then flow through the low impedance path composed by Lf and Cf, rather than their injection into the grid. Hence, the LLCL filter can attenuate the switching frequency harmonics better than the L and the LCL filters while reducing the total inductor value and size [5][6]. The same principle can be repeated by introducing more LC traps designed at other dominant harmonic frequencies (such as 2fsω, 3fsω…), as shown in Fig.1.d [6][7]. Due to the complexity introduced by this filter, the number of trap is kept at one and the selected passive low-pass filter is the fourth order LLCL filter. However, closed loop current control of LLCL-filter-based Grid-connected Converter (LLCL-GcC) may be unstable because of the related resonance problem. Consequently, robust current control strategies are required to maintain system stability.
Li
Li
L2
Cff (R LCL filter
L filter (a) Renewable energy source or load
PCC Lg
Passive low-pass filter
Vdc
Li S(a,b,c)
L2
Digital controller
Li
Cff R Lff R LLCL filter i2(a,b,c)
Vdc
(b) Grid
L2 Cf1
Cfn
Lf1 …
Lfnf
Multi-tuned filter (c)
(d)
Vg(a,b,c)
Fig. 1. (a) Power circuit of the GcC with (a) L filter (b) LCL filter (c) LLCL filter (d) multi-tuned filter
Several standard control strategies with constant switching frequency have been developed in literature. The most used ones are the voltage-oriented PI control (VOC) [8][9], the Direct Power Control associated to the Space Vector Modulator (DPC-SVM) [10] and the Indirect Sliding Mode Power Control (ISMPC) [11][12]. Compared to the VOC and the DPC-SVM control strategies, the ISMPC is characterized by low current Total Harmonic Distortion (THD) factor, fast transient response, non-use of integral terms as well as robustness against external disturbances and parameters variations [12]. However, due to the inherent resonance phenomenon of LLCL filters, the stability region of the ISMPC is limited. Similarly to the LCL filter, passive damping method (PD), achieved by introducing a resistor in series with the filter capacitor, is the simplest solution to damp the LLCL filter resonance [13]. However, this method has several drawbacks, such as reduced system efficiency and power losses [14]. Instead of PD methods, the active damping (AD) ones, obtained by modifying the control algorithm without using dissipative elements, have been also
suggested for LLCL-GcCs. The AD methods are based either on a digital filter (such as a Notch filter [15][16], a lead-lag compensator [17] or quite simply a low-pass filter [18]) or a multi-loop control (such as filter capacitor current inner loop [19][20] and Virtual Resistor based Active Damping (VRAD) [21][22]). The filter based AD methods have the benefit of reducing the sensors number, but they are more sensitive to parameters variations and disturbances. It should be noted that, in most current research works, the PD and AD methods for high order filter based GcCs (such as LCLGcCs and LLCL-GcCs ) are associated only to VOC control strategy. Depending on the grid conditions (weak or stiff grid) [23][24] and configuration (low, medium or high voltage lines, wires length…) [25][26], the grid impedance variations can weaken the damping effect and challenge the control of LLCL-GcCs in terms of stability. In this context, the aim of this paper is to propose an Indirect Sliding Mode Power Control associated to Virtual Resistor based Active
3
Damping method (ISMPC-VRAD) for the LLCL-GcC. The LLCL filter design parameters as well as the ISMPC-VRAD gains are carefully tuned in order to ensure stable operation under severe grid inductance variations while taking into account the influence of LLCL filter parameters changes on the system stability. To summarize, compared to previous related works, the new features of this paper are: 1) The association between robust control strategy (ISMPC) and robust active damping method (VRAD); 2) robustness against large grid inductance variations, overestimated to 13 mH; 3) robustness against filter parameters variations, overestimated to ±20%; 4) a simple and systematic method for the tuning of the ISMPC-VRAD gains in order to ensure simultaneously a low harmonic attenuation rate δ and a grid current THD value less than 5%. This paper is organized as follows. Firstly, in section 2, the mathematical model and the parameters design of the LLCL-GcC are presented. Then, in section 3, the stability analysis of the ISMPC-VRAD for LLCL-GcC is detailed and discussed. Then, in section 4, the theoretical analysis of Section 3 was verified through simulation results achieved under Matlab-Simulink software tool. The obtained simulation results prove performances, effectiveness and robustness of the proposed ISMPC-VRAD algorithm as well as high filtering performances of the designed LLCL-GcC. 2. LLCL filter mathematical model and parameters design 2.1. Mathematical model Assuming that the grid three phases system is balanced, the equivalent single phase representation of an LLCL-GcC power circuit is given by Fig.2. L2g
Vi
1 Lis
+ Vc
Lfs +
Vg + -
1 Cfs
ii ic
+ -
1 L2gs
i2
Fig. 3. Simplified dq-axis LLCL-GcC block diagram
Assuming that the grid voltage is an ideal sine wave, the high frequency LLCL-GcC transfer function F1 is expressed by equation (2). Based on this equation, the LLCL-GcC resonance frequency (that corresponds to zero impedance) is given by (3). The transfer function between iin and Vin (for high frequencies) can be approximated as in (4) [27]. Based on equations (2) and (4), the high frequency transfer function between the grid and the converter currents is expressed by equation (5). At the switching frequency, the previous equation becomes equal to (6). C f Lf s 2 1 i2 n F1 (2) Vin C f ( L f ( Li Lg2 ) Li L2g ) s3 ( Li Lg2 )s
f res
1 2
Li Lg2 L L C f ( Li L2g ) L f C f
(3)
g i 2
iin 1 Vin Li s
(4)
Li (C f Lf s 2 1) i2 n iin C f ( Lf ( Li Lg2 ) Li Lg2 )s 2 ( Li L2g )
(5)
Li (C f L f s2 1) i2 s iis C f ( L f ( Li Lg2 ) Li Lg2 )s2 ( Li Lg2 )
(6)
Bode Diagram GcC
ii
L2
Li Cf
Vdc
Vi
Lf
i2
PCC
Lg
100 LLCL filter resonance
ic
LCL filter resonance
0 Vc
Vg -100
LLCL Filter
Grid utility -200
Fig. 2. Single phase equivalent circuit of LLCL-GcC
s
1 Lf C f
-300
According to this figure, the LLCL filter equations can be -90 fsω=10kHz expressed as follows -135 Vc Vg Vi Vc LLCL filter ii ( a) i2 g (b) -180 LCL filter Li s L2 s (1) -225 1 Vc ( L f s)ic (c) ic ii i2 (d ) -270 Cf s 102 103 104 105 Frequency (Hz) By applying the abc-to-dq coordinate transformation to equations given by (1) and neglecting the decoupling terms Fig. 4. Bode diagram of F1 in case of LLCL and LCL filters between the d and q axes, the simplified block diagram of an Fig.4 shows the Bode diagram of the transfer function F1 of LLCL-GcC power circuit in the dq frame (where the d-axis is both the LLCL filter and the LCL filter (F1 with Lf is set to linked to the grid voltage vector) is depicted on Fig.3. zero). In this figure, the parameters of the LLCL and LCL filters are the same except for the Lf. This figure shows that the LCL and LLCL filters have the same frequency response
4
in the frequency range lower than the filter resonance frequency. Although, it exists a low impedance in the LLCL filter at the switching frequency. This negative impedance is caused by the LC branch and offers to the LLCL filter higher harmonic attenuation at the switching frequency compared to the LCL filter. 2.2. Parameters design
- Step 4: Tuning of Limin The main objective of the converter side inductor Li is to reduce the converter current ripple. The minimum value of Li is given by equation (12.a) [27]. On the other hand, the converter side current ii must verify equation (12.b) in order to avoid inductor saturation problems. Consequently, according to equations (12.a) and (12.b), Limin can be deduced based on equation (13).
The design of an LLCL filter requires the following input Vdc i data: Ugn, Pn, fg, fsω, Vdc and Isat. The tuning of the LLCL Li min (a) Ii max max I sat (b) (12) 6 f s imax 2 filter parameters is detailed in the following steps. - Step 1: Determination of fres condition Vdc Li min (13) Since the resonance frequency is a decreasing function of Lg 12 f s I sat Ii max variable (equation (3)), the range of resonance frequency Where Iimax=I2max (for high frequencies). variation is given by the following equation - Step 5: Tuning of Lf fres min f res ( Lg ) f res max In order to ensure zero impedance at the switching frequency, the value of the inductor in series with the filter Li Lg2max 1 fres min g g capacitor Lf is computed according to the following equation 2 Li L2max C f ( Li L2 max ) L f C f (7) 1 Lf (14) Li Lg2min 1 C f (2 f s ) 2 fres max 2 Li Lg2min C f ( Li Lg2 min ) L f C f - Step 6: Tuning of L 2
On the other hand and in order to avoid resonance problems, fres must be higher than 10 times fg and less than half of fsω [9]. Consequently, in order to avoid system instability due to large grid impedance changes and resonance problems, the range of resonance frequency variation must verify the following condition 10 f g f res min f res ( Lg ) f res max f s / 2 (8) - Step 2: Tuning of LTmax In order to reduce the voltage drops and the losses in the filter, the LT value should be as small as possible. To this purpose, LTmax should be lower than 0.1 pu as shown in equation (9) [27].
LT max 10%
U gn2 2 f g Pn
(9)
- Step 3: Tuning of Cfmax Reactive power consumed by the filter capacitor Qc should be constrained less than λ% of the rated power Pn in order to avoid power factor decrease as shown in equation (10). Based on this condition and for λ equal to 5 [9], the maximum filter capacitor value is given by (11). It should be noted that when the value of Cf is too low, the inductor values must be too high. Otherwise, if the value of Cf is too high, the inductor values will be smaller and therefore the current ripples become more important. To this purpose, it is advisable to begin with a capacitor value equal to one half of the maximum value. If some of the constraints are not verified, the capacitor value should be increased up to the maximum value. 2 Qc U gn C f g % Pn (10) C f max 5%
Pn 2 2 f gU gn
(11)
The main objective of the grid side inductor L2 is to reduce the grid current harmonics according to grid code requirements. Based on the IEEE 519-1992 standard, the value of the grid current THD should be less than 5% [28]. The relation between L2 and Li is given by equations (15.a). L2 aLi (a) where 0 a amax L (15) and amax T max 1 (b ) Li By substituting L2 by its expression given by equation (15.a), equation (6) becomes equal to equation (16) (for Lg equal to zero). In equation (16), δ is the harmonic attenuation rate. It is the ratio between the converter and the grid currents at the switching frequency. The positive solution of equation (16) is given by equation (17). Based on equations (15.a) and (17), L2 can be expressed as in equation (18). C f Li b s2 1 i 2 s (16) iis (1 a )(C f Li bs2 ) C f Li a s2 a
a2 a2 a3
(17)
Li a2 (1 ) (18) a3 Where a2=1+CfLfωsω2 and a3=1+Cfωsω2(Lf+Li ). By substituting L2 by its expression given by equation (18), fresmin and fresmax (expressed by equation (7)) are given by equations (19) and (20), respectively. 1 a4 Li a2 f res min (19) 2 a5 a0 a2 L2
f res max
1 2
b4 Li a2 b5 a0 a2
(20)
5
Where a1=LiLgminCf+LfCf(Li+Lgmin), a4=a3(Li+Lgmin)-a2Li , a5=a1a3-a2a0, b1=Li LgmaxCf+LfCf(Li+Lgmax), b5=b1a3-a2a0, b4=a3(Li+Lgmax)-a2Li and a0=LiCf(Li+Lf). According to equations (8), (19) and (20), δ must verify the conditions expressed respectively by equations (21) and (22) in order to ensure system stability even for large grid impedance variations and resonance problems. On the other hand, the desired harmonic attenuation rate must be greater than a minimum harmonic attenuation rate δmin expressed by equation (23). 10 f g f res min
fres max
1 2
a4 Li a2 a5 a0 a2
Rd
The LLCL filter parameters design has been applied to a system with Ugn equal to 400V, Pn equal to 4kW, fg equal to 50Hz, fsω equal to 10kHz, Vdc equal to 600V and Isat equal to 12A. Based on the design methodology presented and detailed in the previous paragraph, the current harmonic attenuation rate δ should be between 3.3% and 34%. A current harmonic attenuation rate δ equal to 7% is selected. Tab.1 summarizes the used system and the LLCL filter parameters. Table 1. System and LLCL filter parameters
b4 Li a2 f s / 2 b5 a0 a2
a a ( f s ) 2 a2 Li 0 2 b4 b5 ( f s ) 2
min
(25)
n
a5 (20 f g ) 2 a4
1 2
(24)
Pd 3 Rd (iin i2n ) 2
(21)
a2 Li a0 a2 (20 f g ) 2
( Z LC ) res 1 1 (2 f res L f ) 2 f res C f
(22)
System
(23)
LLCL filter
C f L f s2 1 (1 amax )(C f L f s2 ) C f Li amaxs2
Parameter Ugn Pn fsω fg Vdc Li L2 Cf Lf Rd Lg
Damping resistor Grid inductance
The desired harmonic attenuation rate δ should be chosen according to equations (21), (22) and (23). Moreover, it shouldn’t be too high since the harmonics are lower when δ is lower. In other words, when δ is lower, the obtained grid current THD value is lower. The selection of δ allows the determination of a based on equation (17). Then, the L2 value is deduced according to equation (18). - Step 7: Tuning of Rd In order to avoid resonance problems, the simplest solution is to add a resistor in series with the filter capacitor. The added damping resistor Rd is computed according to equation (24). The power losses Pd associated to Rd are given by equation (25) [9]. In equation (25), λ is a positive constant selected so that the power losses related to the damping resistor do not exceed 1% of the rated active power.
Value 400V 4kW 10kHz 50Hz 600V 5mH 2mH 4µF 63.33µH 6Ω Lgmin=0mH and Lgmax=13mH
3. ISMPC-VRAD for LLCL-GcC The performance of an LLCL-GcC depends not only on an appropriate design methodology of the filter, but also on an effective control strategy. This is due to the fact that the LLCL filter resonance phenomenon and the large changes in the grid inductance may compromise the system stability. Fig.4 shows the proposed robust ISMPC-VRAD for LLCLGcC. It is made up of both an internal and external control loop. The internal one is based on the ISMPC-VRAD algorithm, while the external one is based on a PI controller that controls the dynamic and the shape of the dc-link voltage Vdc. ISMPC-VRAD
Vdc*
+ Vdc
i2d* + -
PI
dc-link voltage control loop
Si2d
θdq
abc-to-dq i2q* +
-
i2q
i2d
i2(a,b,c) i2q
Vgd -ωLT
+ +
Vi(d,q)eq
i2(d,q) i2d
Videq Viqeq
ωLT
Si2q Si2(d,q)
K(d,q) +
Vg(a,b,c) abc-to-αβ & tang-1
Sgn
Vg(a,b,c) θdq
abc-to-dq
Vg(d,q)
VRAD Vi(d,q)* +
+
Vi(d,q)
*
LT
Vi(d,q)*
Vi(d,q) + - ic(d,q)
+
θdq
Kr
Vi(d,q)att
Q(d,q)
Grid synchronization
*
ic(a,b,c) θdq
abc-to-dq
dq-to-abc Vi(a,b,c)*
Vdc
SVM
ISMPC S(a,b,c)
Fig. 5. ISMPC-VRAD for LLCL-GcC
The internal loop incorporates the ISMPC and the
VRAD algorithms. The ISMPC algorithm controls the grid
6
currents in the dq synchronous reference frame. The d-axis grid current reference i2d* is computed by the PI controller of the outer dc-link voltage control loop, whereas the q-axis grid current i2q* is set to zero in order to impose a unit power factor operation. The VRAD algorithm is used to actively damp the LLCL filter resonance. As shown in this figure, it is achieved by sensing the capacitor current, multiplying it with a constant Kr and subtracting the result from the dq components Vi(d,q)* of the output power converter voltage vector. For such case, the constant Kr behaves like a real damping resistor in series with the filter capacitor without supplementary losses and encumbrance. Finally, the switching states S(a,b,c) of the converter are generated based on the SVM module. Note that the Park transformation (abc-to-dq) and the inverse Park transformation (dq-to-abc) are based on the position θdq of the grid voltage vector. The principle of the ISMPC algorithm is presented in the next paragraph. 3.1. Principle of ISMPC algorithm
two sliding surfaces (Si2d=0) and (Si2q=0). S i 2 d i2*d i2 d S i 2 q i i2 q
(29) The ISMPC algorithm is synthesized so that the switching functions are attracted to their sliding surfaces during transient state (attractive mode). This means that the d and q grid side current components will be also attracted to their references. For this purpose, the switching functions (Si2d and Si2q) and their time derivatives must verify the following attractive conditions. Si 2 d S i 2 d 0 (30) S S 0 (31) i2q
i2q
Once the sliding surfaces are reached, and in order to keep the switching functions on their sliding surfaces during steady state (sliding mode), the invariance conditions given by equations (32) and (33) must be verified. S i 2 d 0 and Si 2 d 0 (32) S 0 and S 0 (33) i2q
For fundamental signals, the LLCL filter can be approximated to an inductor with a value LT equal to the sum of the two inductor values Li and L2 [27]. Based on Fig.2 (for fundamental signals), the LLCL-GcC filter equations in the dq reference frame are given by equations (26) and (27). di Vid LT 2 d g LT i2q Vgd (26) dt di2 q (27) Viq LT g LT i2 d dt In case of ISMPC, the trajectory of the grid currents is made up of two modes as shown in Fig.6. The first one is the attractive mode. During this stage, the trajectory starts from a zero initial point and moves until it reaches the sliding surface at t0 (during transient state). The second mode is the sliding mode. During this stage, the trajectory remains on the sliding surface (during steady state). i2dq
i2dq*
(28)
* 2q
i 2q
The ISMPC-VRAD algorithm is executed at each sampling period Te and between two consecutive sampling periods, the references i2d* and i2q* are constant. So, their time derivatives are null (di2d*/dt=0 and di2q*/dt=0). Consequently, based on equations (26), (27), (28) and (29), the time derivatives of the switching functions are expressed as follows di 1 Si 2d i 2 d (Vgd g LT ii 2 q Vid ) (34) dt LT dii 2 q
1 ( g LT ii 2 d Viq ) (35) dt LT The computation of the reference voltage vector in the dq synchronous reference frame must be done so that the attractive and invariance conditions are satisfied. The reference voltage vector is composed of two terms as shown in equations (36) and (37). The first one (Vidatt and Viqatt) ensures the control of the system during the attractive mode and it is active in the transient state. The second one (Videq and Viqeq) ensures the control of the system during the sliding mode and it is active in the steady state. Vid* Vidatt Videq (36) Si 2 q
Viq* Viqatt Viqeq (37) Based on equations (28), (29), (34) and (35), Videq and Viqeq are deduced as in (38) and (39), respectively.
Sliding mode Attractive mode t t0
Fig. 6. Trajectory characterizing the ISMPC
The objective of the ISMPC is to control the active and reactive power through the control of i2d and i2q, respectively. To this purpose, Si2d and Si2q are defined as the difference between i2d* (respectively i2q*), and i2d (respectively i2q) as shown in equations (28) and (29), respectively. These switching functions Si2d and Si2q define
i2 d * i2 d 1 eq S i 2 d L (Vgd g LT i2 q Vid ) 0 T Videq Vgd g LT i2 q
(38)
i2 q * ii 2 q 1 eq Si 2 q L ( g LT i2 d Viq ) 0 T eq Viq g LT i2 d
(39)
S i 2 d 0 S i 2 d 0
S i 2 q 0 S i 2 q 0
7
According to equations (34), (35), (36), (37), (38) and (39), Vid* and Viq* are expressed as follows dS Vid* LT i 2 d Vgd LT i2 q dt V eq Vidatt
(40)
id
dSi 2q Viq* LT LT i2 d dt eq V
(41)
iq
Viqatt
Equations (40) and (41) show that Vidatt and Viqatt include the time derivatives of Si2d and Si2q. The selection of a constant velocity and a proportional action attractive structure [11], allows the deduction of Vidatt and Viqatt, which are given by the following equations Vidatt
dS LT i 2 d LT (Qd sgn( Si 2d ) K d Si 2 d ) dt
att iq
V V
eq iq
3.2. Choice of Kr The main goal of the VRAD method is to eliminate the real damping resistor Rd and its associated power losses. Fig.7.a and Fig.8 respectively show the un-damped and damped high frequency equivalent single phase representation of an LLCL filter without considering the grid impedance. Li
L2 Cf
Vi
Lf
Vi
H1(s)
ic
H2(s)
V’ c
i2
ic
Vg=0 + Vi -
V’ c
Un-damped LLCL Filter
(a)
Kr
(b)
H1(s)
V’ c
ic
(c)
H2(s)
ic Vc
Vg=0
Damped LLCL Filter
Fig. 8. High frequency single phase circuit with damping resistor in series with the filter capacitor
H1
L2 (1 C f L f s 2 ) Vc ' Vi C f ( Li L2 Li L f L2 L f )s 2 Li L2
(46)
H2
Cf s ic Vc' 1 C f L f s 2
(47)
(45)
The tuning of the K (K=Kd=Kq) and Q (Q=Qd=Qq) constants as well as the virtual resistor Kr is detailed in the following steps.
ii
i2
Rd
V
LT ( Qq sgn( S i 2 q ) K q S i 2 q ) g LT i2 d
L2
Lf
Vi
(44)
LT ( Qd sgn( S i 2 d ) K d Si 2 d ) Vgd g LT i2 q
Li Cf
dSi 2 q
Vid* Vidatt Videq * iq
ii
(42)
(43) LT (Qq sgn( Si 2q ) K d Si 2q ) dt Where Qd, Kd, Qq and Kq are positive constants. Based on equations (40), (41), (42) and (43), Vid* and Viq* are expressed as follows
Viqatt LT
VRAD method can be presented by Fig.7.c. Based on this figure, the transfer function H3 of the modified system is given by equation (48). When passive damping is realized by adding a resistor Rd in series with the filter capacitor (Fig.8), the transfer function G1 of the whole system is given by equation (49). In order to obtain the same poles for the system with VRAD method and the one with a real damping resistor Rd, the denominators of H3 and G1 must be equal. As a result, the Kr value can be deduced according to equation (50). Thus, by considering the selected damping resistor Rd in Section 2.2 (Rd=6Ω), the corresponding Kr value deduced from equation (50) is equal to 18.
Fig. 7. High frequency single phase circuit (a) without damping resistor (b) block diagram of the LLCL filter (c) modified control structure
For the circuit given by Fig.7.a, the transfer function between Vc’ and Vi is expressed by equation (46), while the transfer function between ic and V’c is expressed by equation (47). Based on these equations, the un-damped high frequency LLCL filter model is given by Fig.7.b. According to Fig.5 and Fig.7.b, the control structure of the
H3 G1
L2 (1 C f L f s 2 ) C f ( Li L2 Li L f L2 L f ) s 2 C f L2 K r s Li L2 L2 (1 C f L f C f Rd s ) C f ( Li L2 Li L f L2 L f ) s 2 C f Rd (Li Li )s Li L2
K r Rd
Li L2 L2
(48) (49) (50)
3.3. Choice of K and Q Fig.9 and Fig.10 respectively show the harmonic attenuation rate δ and the grid current THD with regard to K and Q values. It can be noted, based on these figures, that when K and Q increase, the harmonic attenuation rate δ and the grid current THD increase. According to Fig.9, the (K,Q) couples that ensure a harmonic attenuation rate equal to 7% (value selected by the designer during the LLCL filter design) are (150,300), (200,350), (350,350), (300,250), (400,200), (400,250) and (350,100). Based on Fig.10, all the previously selected (K,Q) couples ensure a grid current THD value below than 5%. This is due to the fact that the chosen harmonic attenuation rate δ is very low (7%). For the selected (K,Q) couples, the corresponding grid current THD values are 2.2%, 3.2%, 3.9%, 2.6%, 2.2%, 2.18% and 1.3%, respectively.
8 δ%
(K,Q)=(350,350) δ=7%
δ (%)
7
(K,Q)=(300,250) δ=7%
(K,Q)=(200,350) δ=7%
(K,Q)=(400,200) δ=7%
67 (K,Q)=(150,300) δ=7%
(K,Q)=(400,150) δ=7%
3.3 Constrained by 3.3 < δ < 34
34 (K,Q)=(350,100) δ=7%
7 Q 500
K
400 400 300300 200200 100100
500 400 400 300 300 00 0
100 100
200 200
7
(K,Q)=(200,350) THD=3.2%
(K,Q)=(400,200) (K,Q)=(400,150) (K,Q)=(350,100)
0 Lgmin=0
Lg(mH) Lgmax=13
Fig. 11. Harmonic attenuation rate δ according to grid impedance Lg variation for different values of (K,Q) THD (%) Constrained by THDi2 < 5%
Fig. 9. Harmonic attenuation rate δ according to (K,Q)
THD(%)
(K,Q)=(350,350) (K,Q)=(150,300) (K,Q)=(200,350) (K,Q)=(300,250)
5
(K,Q)=(350,350) THD=3.9%
(K,Q)=(350,350) (K,Q)=(150,300) (K,Q)=(200,350) (K,Q)=(300,250) (K,Q)=(400,200) (K,Q)=(400,150) (K,Q)=(350,100)
(K,Q)=(300,250) THD=2.6%
(K,Q)=(150,300) THD=2.2%
(K,Q)=(400,200) THD=2.2% (K,Q)=(400,150) THD=2.18%
5
(K,Q)=(350,100) THD=1.3%
Lg(mH)
0 Lgmin=0
Lgmax=13
Fig. 12. Grid current THD according to grid impedance Lg variation for different values of (K,Q)
Q K 500 400400 300 300 200 200 100 100
0 0
100 100
200 200
300 300
400 400
500
Fig. 10. Grid current THD according to (K,Q)
Fig.11 and Fig.12 respectively show the harmonic attenuation rate δ and the grid current THD value with regard to the grid impedance change for the obtained (K,Q) couples. In order to avoid system instability due to resonance problems and large grid impedance variations, δ must be between 3.3% and 34% and in order to meet new grid codes and power quality requirements, the grid current THD value must be below 5%. It can be noted, according to Fig.12, that the grid current THD is lower than 5% for the different values of (K,Q) and despite the high grid impedance variation. However, when Lg increases, the harmonic attenuation rate δ is decreased for a large set of (K,Q) couples as shown in Fig.11. The (K,Q) couples that ensure at the same time an harmonic attenuation rate δ between 3.3% and 34% and a grid current THD value lower than 5%, even for a large grid impedance variation are (350,350), (150,300) and (200,350). A value of (150,300) was chosen for the (K,Q) couple.
Moreover, the robustness of the system against filter parameters variations was investigated for the selected (K,Q) couple. Tab.2 shows the harmonic attenuation rate δ and the grid current THD for Lg equal to 13mH, Cf varies from 1.6μF to 2.4μF (2μF±20%), Li varies from 4mH to 6mH (5mH±20%), Lf varies from 50.66µH to 75.99µH (63.33µH±20%) and L2 varies from 1.6mH to 2.4mH (2mH±20%). It is found that for all these cases, the harmonic attenuation rate δ is larger than 3.3% and the grid current THD is below 5%. So, the system stability is ensured under large variations of grid impedance and LLCL filter parameters. Table 2. Harmonic attenuation rate δ and grid current THD value for large grid impedance variation and LLCL filter parameters variation Lg=13mH Li ± 20% (mH) Cf ± 20% (µF) Lf ± 20% (µH) L2 ± 20% (mH)
4 6 1.6 2.4 50.66 75.99 1.6 2.4
δ (%) 4.3 4.8 3.6 4.1 3.45 3.55 3.42 3.4
THD (%) 1.4 1.7 0.89 0.94 0.76 0.78 0.73 0.6
9
The obtained controller parameters are Kr=18, K=150 and Q=300. The ISMPC-VRAD algorithm was tested based on Matlab-Simulink software tool. During simulation tests, the switching frequency is equal to 10kHz and the dc-link capacitor is firstly charged to 100V. Fig.13.a and Fig.13.b present the responses of the dc-link voltage Vdc and the grid
current i2a before and after connecting a resistive load Rch=80Ω in the DC side at 0.25s. It can be noted that the Vdc voltage is controlled with good accuracy during steady state operation. Fig.13.c shows the waveform of the grid current i2a with regard to the grid voltage Vga during steady state operation. Vga
Vgmax
150V
Lg=0mH Rch=80Ω
i2a
4A
Vdc
100V
Load connection Rch=80Ω
-4A -Vgmax
(a) 0
0.125
0.25
0.375
0.5
(c)
0.06
0.07
0.08
Rch=80Ω
0A
0.1
0.11
0.12
Lg=13mH
i2a
-4A
4A
0.09
0A
i2 -4A
-4A
(b) 0
0.125
0.25
0.375
(d)
THD(i2a)=1.1% 0
0.5
0.125
0.25
(g) Lg=0mH Rch=80Ω
7.3% 4A
0A
190
0.5
(h)
Lg=0mH Rch=80Ω
4A
0.375
195
200
205
210
-4A
0.52%
0A
190
195
200
205
210
-4A
(e)
THD(iia)=9% 0.09 0.11
0.07
(f) 0.13
0.07
0.15
0.09
4A
-4A
0A
0A
-4A
0.11
0.13
i2a
Lg=0mH Rch=80Ω
i2a
THD(i2a)=0.8% 0.15 Lg=13mH Rch=80Ω
-4A THD(i2a)=0.45% Low damped system
0
VRAD method
0.2
(i) 0.4
Low damped system
0
VRAD method
0.2
(j) 0.4
Fig. 13. (a) dc-link voltage Vdc response (b) grid current i2a response (c) grid voltage Vga and current i2a waveforms at steady state operation (d) grid current i2a response for Lg=13mH (e) power converter current iia response (f) grid current i2a response (g) high frequency spectra of iia (h) high frequency spectra of i2a (i) i2a before and after enabling the VRAD method at 0.2s for Lg=0mH (a) i2a before and after enabling the VRAD method at 0.2s for Lg=13mH
Based on this figure, a unit power factor operation was obtained as expected. Fig.13.e and Fig.13.f show the simulation results of the converter current iia and grid current i2a, respectively. It can be noted that the current harmonic components are almost disappeared at the switching frequency. The THD of the simulated converter current is equal to 9%, while the one of the simulated grid
current is equal to 0.8%. Fig.13.g and Fig.13.h respectively present the high frequency spectra of the simulated converter and grid currents. Based on these figures, the switching frequency current harmonic component on the converter side iisω is equal to 7.3% and the one on the grid side i2sω is equal to 0.52%. Thus, the harmonic attenuation rate δ is well equal to 7%. In order to test the robustness of
10
the ISMPC-VRAD, additional inductors of 13mH are inserted in series with the LLCL filter grid side inductor. As shown in Fig.8.d, the system remains stable despite of a large variation of the grid inductor value. Fig.13.i and Fig.13.j shows the grid current of a low damped system and the one of a damped system through VRAD method for Lg=0mH and Lg=13mH, respectively. According to these figures, when the system is low damped, the resonance ripples are clearly increased and the system is close to the instability region. By enabling the active damping at 0.2s, the resonance ripples are damped out and the system becomes more stable despite of a large variation of the grid inductor value. Simulation results indicate the effectiveness and the robustness of both ISMPC-VRAD algorithm and the designed LLCL-GcC. Finally, Tab.3 shows a comparison
between the ISMPC-VRAD for LLCL-GcC and the VOCVRAD for LCL-GcC [24]. In this comparison, the LCL and LLCL filters parameters are the same (expect for LCL filter, the Lf value is equal to 0mH). This table shows that the ISMPC-VRAD for LLCL-GcC has better performances compared to the VOC-VRAD for LCL-GcC. This is due to the fact that the LLCL filter ensures higher filtering performances at the switching frequency (lower grid current THD value) compared to the LLCL filter thanks to the use of the additional small inductor in series with the filter capacitor. Moreover, the ISMPC-VRAD is characterized by faster transient response thanks to the non-use of integral terms, lower grid current THD value as well as more robustness against external disturbances and parameters variations compared to the VOC-VRAD.
Table 3. Comparison between performances of VOC-VRAD for LCL-GcC and ISMPC-VRAD for LLCL-GcC Criteria/Control strategy Dc-link voltage regulation Unity power factor operation Transient response Use of Integral terms Tuning of control parameters Grid current THD value (fsω=10kHz) Stability Grid current THD Robustness against large variation of Lg (fsω=10kHz and Lg=13mH) Stability for filter parameters variation (overestimated to ±20%)
4. Conclusion This paper proposed an ISMPC-VRAD for the LLCLGcCs. To this purpose, the LLCL filter design parameters as well as the ISMPC-VRAD gains have been carefully computed in order to ensure stable operation under severe grid inductance variations (overestimated to 13mH) while taking into account the influence of LLCL filter parameters changes (overestimated to ±20%) on the system stability. The obtained simulation results gave proof of the effectiveness, performances and reliability of the implemented ISMPC-VRAD algorithm as well as the high filtering performances of the used LLCL-GcCs. Acknowledgements “This work was supported by the Tunisian Ministry of High Education and Research under Grant LSE-ENITLR11ES15” References [1] M.A. Elsaharty, H.A. Ashour. Passive L and LCL Filter Design Method for Grid-Connected Inverters, IEEE Conference in innovative Smart grid technologies, pp.13-18, 2014. [2] S. Jayalath, M. Hanif, “Generalized LCL-Filer Design Algorithm for Grid-connected Voltage Source Inverter”, IEEE Trans. Ind. Electron, vol. PP, pp.1537– 1547, 2016. [3] S. Lim, J. Choi et al., “LCL Filter Design for Grid
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+
++
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